Post-necking Hardening Research Articles

Influence of specimen width on the elongation-to-fracture in cyclic-bending-under-tension of commercially pure titanium sheets

In recent publications, we have shown that repeated bending and unbending during tension can significantly increase the percent elongation-to-fracture (ETF) of metallic sheets. A custom-built machine for cyclic bending under tension (CBT) testing was designed and used to test several sheet metals. In particular, the improvements in ETF using CBT were over 3× relative to simple tension (ST) for commercially pure titanium (cp-Ti) sheets. The present paper evaluates the influence of specimen width on the achieved ETF in CBT of cp-Ti sheets. To facilitate the study, the CBT machine was furnished with wide grips to enable processing of sheets in addition to testing narrow strips. The ETF was observed to reduce with the sheet width. To rationalize the observations, the strain rate-independent plasticity theory of von Mises (J2) with isotropic expansion of yield surfaces of cp-Ti was used in finite elements (FE) to perform a set of simulations. To facilitate the FE simulations of the CBT process stretching the sheets of cp-Ti far beyond the point of maximum uniform strain in ST, the post-necking hardening behavior was inferred along RD, TD, and 45 sheet directions using an established methodology combining ST tests, CBT tests, and modeling material behavior during CBT. Simulated geometries and mechanical fields were found to be in good agreement with corresponding measurements for every specimen width. While predicted axial strains were similar for every width, the strain path shifted from uniaxial tension for the narrow specimen toward plane-strain tension for the wider specimens. As such, as plane-strain tension was approached via progressively wider specimens, the width strains reduced, while the thinning strains increased. The increase in the thinning strain was established as the primary reason for the reduced ETF with increasing sheet width. Experimental and FE simulation results along with the insights into the influence of specimen width on the ETF in CBT of cp-Ti are presented and discussed.

Using surround DIC to extract true stress–strain curve from uniaxial tension experiments

The exact stress–strain curve can be directly identified from uniaxial tension experiments up to the point of onset of diffuse necking. To gain insight into the post-necking hardening response of metals, other experimental methods such as the bulge, compression or torsion experiments are typically employed. Here, we introduce the idea of using surround DIC to obtain accurate specimen shape measurements of tensile specimens with rectangular gage section. Based on the results from a series of detailed three-dimensional finite element simulations of ASTM E8 type of uniaxial tension experiments on a wide spectrum of steel and aluminum behaviors, it is proposed to combine the history of the axial strain on the surface at the specimen center with the average axial stress to extract the stress–strain curve for strains of up to axial true strain of 1. The estimation uncertainty of this stress–strain curve estimation procedure is of the same order as that of the associated experimental uncertainties. This result is validated through an additional computational study for more than 100 distinct hardening behaviors. Furthermore, a surround DIC system composed of four stereo DIC systems is built and used to determine the stress–strain curve for a 1.5 mm thick DP780 steel. In addition, bulge experiments are performed on the same material revealing a good agreement of the post-necking hardening behavior determined through uniaxial and bulge testing.

Stress-state-dependency of post-necking hardening rule and its influence on ductile fracture prediction

This paper presents the investigation of the stress-state-dependency of the post-necking hardening rule of the Q355B structural steel and its influence on fracture prediction. The comparisons of the testing result and the simulation based on the Mises model with the stress-state-independent hardening rule indicate that triaxiality dependency under axisymmetric tension is negligible, while the reduced yield stress increases with the rising triaxiality under plane strain. In addition, the overestimated predicted force based on the Mises model is continuously ramping up along with the plastic deformation under the shear-dominant condition. It can be considered by the reduced post-necking hardening rate under the requirement of convexity of the yield surface. The stress-state-dependent hardening model (SDH model) is herein proposed including the modified triaxiality-dependent Hosford yield criterion and state-dependent post-necking hardening rule. The comparative studies of the proposed and conventional models show similar calculated fracture strains under high triaxiality, but considerably larger fracture strain under low triaxiality from the proposed SDH model, since the reduced post-necking hardening rate intensifies the localization of plastic deformation. Using the proposed pressure-dependent maximum shear stress (PMSS) fracture model, the calibrated fracture parameters based on the data from SDH the model produce better fracture simulations under various stress states, especially the complex condition in multi-hole plate (MHP) specimen. It emphasizes the necessity of the consideration of stress-state-dependency in the post-necking hardening rule for accurate fracture prediction.

Inverse Finite Element Approach to Identify the Post-Necking Hardening Behavior of Polyamide 12 under Uniaxial Tension.

Finite-element (FE) simulations that go beyond the linear elastic limit of materials can aid the development of polymeric products such as stretch blow molded angioplasty balloons. The FE model requires the input of an appropriate elastoplastic material model. Up to the onset of necking, the identification of the hardening curve is well established. Subsequently, additional information such as the cross-section and the triaxial stress state inside the specimen is required. The present study aims to inversely identify the post-necking hardening behavior of the semi-crystalline polymer polyamide 12 (PA12) at different temperatures. Our approach uses structural FE simulations of a dog-bone tensile specimen in LS-DYNA with mesh sizes of 1 mm and 2 mm, respectively. The FE simulations are coupled with an optimization routine defined in LS-OPT to identify material properties matching the experimental behavior. A Von Mises yield criterion coupled with a user-defined hardening curve (HC) were considered. Up to the beginning of necking, the Hockett–Sherby hardening law achieved the best fit to the experimental HC. To fit the entire HC until fracture, an extension of the Hockett–Sherby law with power-law functions achieved an excellent fit. Comparing the simulation and the experiment, the following coefficient of determination could be achieved: Group I: > 0.9743; Group II: > 0.9653; Group III: > 0.9927. Using an inverse approach, we were able to determine the deformation behavior of PA12 under uniaxial tension for different temperatures and mathematically describe the HC.

Hybrid experimental-numerical strategy for efficiently and accurately identifying post-necking hardening and ductility diagram parameters

Although various effective models for simulating ductile fracture in metallic materials have been established, the methods to identify the required parameters have scarcely been clarified. The difficulty for parameters identification of ductile fracture models is the actual bottleneck for accurate numerical predictions in ductile fracture simulation. In this paper, a strategy to efficiently identify the post-necking strain behaviour and the ductility diagram parameters is proposed by a hybrid experimental-numerical procedure. In the proposed strategy, only two types of experiments are required: a conventional tensile test and a crack growth test. All the parameters to simulate the post-necking strain hardening behaviour and the ductile fracture can be uniquely identified via numerical optimisations using load-displacement curves of the two tests. The proposed strategy was applied to an A2024T3 aluminium alloy and the post-necking hardening and ductility diagram parameters were identified by a flat tensile test and an Arcan test. Two types of plate specimens with a hole were then used to validate the proposed strategy. Good agreements between the experimental results and the numerical predictions clearly show the feasibility and accuracy of the proposed strategy. The results demonstrated that the proposed strategy is simple but effective. Therefore, it can be a general basis for characterising mechanical properties of various metallic materials.

Plastic deformation of AA6061-T6 at elevated temperatures: Experiments and modeling

The temperature-dependent work-hardening of AA6061-T6 aluminum alloy is investigated using a combined experimental-numerical approach. Uniaxial tension experiments are performed from room temperature to 500 °C, and the force-displacement curves up to fracture are established. Both the yield strength and the work-hardening rate of the material decrease with increasing temperature. Despite the marked increase in the elongation-to-fracture, the uniform elongation remains limited, indicating the early appearance of diffuse necking in the elevated temperature tests. A simplified technique for identifying the work-hardening curves at elevated temperatures and beyond the limit of uniform deformation in uniaxial tension is proposed. The technique uses an isothermal, rate-dependent, finite element (FE) model of the experiments. The post-necking hardening curve is represented by a hybrid Swift/Voce hardening law combined with a form of Johnson-Cook strain-rate dependence. To identify the material model parameters at each temperature, an objective function that correlates the experimental and FE-predicted force-displacement curves is constructed and minimized. These model parameters are then fit with Fourier series across the temperature range considered here, providing smooth functions suitable for subsequent implementation in FE analyses of forming and structural problems. As an illustration, it is shown that a FE model utilizing these material parameters reproduces the full-field features of the elevated temperature experiments very well.

Stepwise modelling method for post necking characterisation of anisotropic sheet metal

Modelling and simulation are important tools during design and development processes. For accurate predictions of, e.g. manufacturing processes or final product performance, reliable material data is needed. Usually, the applied material models are calibrated by utilising direct methods such as conventional uniaxial tensile/compression tests but also inverse methods are occasionally applied. Recently, an effective inverse method, the stepwise modelling method (SMM), was presented. By using SMM, the flow stress from initial yielding, beyond necking to final fracture, can be determined. However, the method is developed for sheet materials having isotropic von Mises hardening. In this paper the SMM is extended for post necking characterisation of anisotropic sheet metals using the Barlat yield 2000 criterion. The novel method was applied to analyse the post necking plasticity of the widely used aluminium alloy AA6016 in T4 condition and the aluminium alloy AA5754 in H111 condition. The latter alloy has reported to show serrated yielding, also known as the Portevin–Le Chatelier effect. The obtained flow stress curves agree well with the curves form conventional uniaxial tensile tests up to the point of necking and show credible post necking predictions to final fracture. Furthermore, SMM showed that it could handle the effect of serrated yielding for AA5754-H111. Hence, the novel approach can be used to characterise the post necking hardening of a variety of anisotropic sheet metals and thereby contributes to efficient and reliable material model calibration.

Local and global tensile deformation behavior of AA7075 sheet material at 673oK and different strain rates

The constitutive behavior of AA7075 aluminum sheet like other metallic sheet materials is highly temperature and strain rate dependent at elevated temperatures. The uniaxial tensile test is commonly employed to characterize the temperature and strain rate dependent deformation behavior. However, the traditional uniaxial tensile testing and data analysis procedure treats the gauge length of sample as a uniformly-deforming (or homogeneous) region. This may not be accurate due to early localized deformation that is initiated in the gauge region at elevated temperatures. Treating the post-necking hardening behavior of uniaxial tensile samples has been an important task to expand the effective range of true stress-true strain curves at elevated temperatures for metal and alloys. To explore this issue, uniaxial tensile tests on AA7075-F sheets (with as-fabricated temper) were conducted at 673o K and four different test speeds. The test procedure also utilized an on-line 2-D digital image correlation (DIC) system method to obtain full-field strain map from the deforming gauge region of the test specimen. Local stress-strain curves at different points along the gauge length were then calculated based on the recorded local in-plane principal stress and strain values from macroscopic force data and current cross-section of the specimen at the chosen points. In this manner, the DIC strain analysis not only enabled the determination of stress-strain response but also the determination of strain rates at specific chosen points along the gauge length. The local stress-strain curves revealed significant deviation at an earlier stage of deformation from the global (or macroscopic) stress-strain response based on the entire gauge region. To estimate the constitutive behavior at different strain rates, a correction method to traditional stress-strain curves was applied to obtain a family of stress-strain curves at different constant strain rates based on the local stress-strain curves. Additionally, the material deformation response in the stress, strain and strain rate space in the 3D Cartesian coordinate system was also constructed for tests conducted at different stretching speeds. Further, a verification test was conducted to see if the above methodology could provide an accurate estimation of local deformation behavior of AA7075 sheet. Furthermore, the above experimental method was also coupled with the Gurson-Tvergarrd-Needleman (GTN) ductile void damage model and incorporated into ABAQUS-Explicit general purpose finite element (FE) analysis code via a VUMAT material subroutine. Good agreement was obtained for both the local and global stress-strain curves between the FE test simulation and experimental stress-strain results. The proposed experimental methodology and the resulting stress-strain-strain rate behavior is believed to be applicable in describing the strain rate-dependent constitutive behavior of AA7075-F sheet metals at 673o K.

Tensile deformation behavior of TRIP-aided bainitic ferrite steel in the post-necking strain region

Transformation induced plasticity (TRIP) steels present a remarkable balance of strength and ductility. However, their post-necking hardening behavior, which is required for press-forming and autom...

Experimental and numerical investigation of deformation characteristics during tube spinning

The primary objective of this study is to predict the geometric shape and thickness change during multi-pass tube spinning to a hemispherical shape. A computationally efficient, axisymmetric finite element model of the tube-spinning experiments is described. Uniaxial tensile tests are conducted for the development of the material model. The post-necking hardening curve is identified using a hybrid experimental-numerical method and represented by a combined Swift-Voce model. For validation of the finite element model, spinning experiments are performed at room temperature on thin-walled cylinders of 6061-O aluminum alloy. The experiments are interrupted after spinning passes 2, 4, 6, and 8 and the shape and thickness are measured. Also, local strain measurements on the final spun tube (pass 8) are accomplished by a scribed grid. By comparing the predictions to the experiments, good agreement is obtained on the shape, thickness, and strain evolution in multi-pass spinning. The deformation mechanism of this process is described by analyzing the history of plastic strain on an element of the numerical model.

Inferring Post-Necking Strain Hardening Behavior of Sheets by a Combination of Continuous Bending Under Tension Testing and Finite Element Modeling

This paper presents a combined experimental and simulation approach to identify post-necking hardening behavior of ductile sheet metal. The method is based on matching a measured load-displacement curve from a continuous bending under tension (CBT) test with the curve simulated using the finite element method (FEM), while adjusting an input flow stress curve into the FEM. The CBT test depletes ductility uniformly throughout the gauge section of a tested sheet, and thus, stretches the sheet far beyond the point of maximum uniform strain in a simple tension (ST) test. Having the extended load-displacement curve, the calibrated flow stress curve is extrapolated beyond the point of necking. The method is used to identify the post-necking hardening behavior of an aluminum alloy, AA6022-T4, and two dual-phase (DP) steels, DP 980 and DP 1180. One measured load-displacement curve is used for the identification of a flow stress curve per material, and then the flow stress curve is used to simulate two additional measured load-displacement curves per material for verification. The predictions demonstrate the utility of the developed CBT-FEM methodology for inferring the post-necking strain hardening behavior of sheets. Furthermore, the results for AA6022-T4 are compared with the hydraulic bulge test data. Unlike the hydraulic bulge test, the CBT-FEM method can predict increasing as well as decreasing anisotropic hardening rate in the post-necking regime. The latter is associated with probing stage IV hardening. The proposed methodology along with the results and these key advantages are presented and discussed in this paper.

Evaluation Study on Iterative Inverse Modeling Procedure for Determining Post-Necking Hardening Behavior of Sheet Metal at Elevated Temperature

The identification of the post-necking strain hardening behavior of metal sheet is important for finite element analysis procedures of sheet metal forming process. The inverse modeling method is a practical way to determine the hardening curve to large strains. This study is thus focused on the evaluation of the inverse modeling method using a novel material performance test. In this article, hot uniaxial tensile test of a commercially pure titanium sheet with rectangular section was first conducted. Utilizing the raw data from the tensile test, the post-necking hardening behavior of the material is determined by a FE-based inverse modeling procedure. Then the inverse method is compared with some classical hardening models. In order to further evaluate the applicability of the inverse method, biaxial tensile test at elevated temperatures was performed using a special designed cruciform specimen. The cruciform specimen could guarantee that the maximum equi-biaxial deformation occurs in the center section. By using the inverse modeling procedure, the hardening curves under biaxial stress state are able to be extracted. Finally the stress-strain curves obtained from the two experiments are compared and analysis studies are provided.

Experiments and simulation of shape and thickness evolution in multi-pass tube spinning

The primary objective of this study is to predict the geometric shape and thickness change during multi-pass tube spinning to a hemispherical shape. Interrupted spinning experiments are performed at room temperature on cylinders of 6061-O aluminum alloy. The shape and thickness after 2, 4, 6 and 8 spinning passes are measured. An axisymmetric finite element model of the tube spinning experiments is described. Uniaxial tensile tests are conducted for the development of the material model. The post-necking hardening curve of the material is identified using a hybrid experimental-numerical method. The stress-strain identified in this way is fitted to a combined Swift-Voce model. The results of the simulation are compared to the interrupted spinning experiments. Good agreement is obtained on the shape evolution in multi-pass spinning. The effect of roller paths on shape and thickness change during multi-pass tube spinning targeting a hemispherical shape with uniform thickness were predicted by axisymmetric modeling and validated with experimental results.

Ductile fracture of AA6111 alloy including the effect of bake-hardening

The thermal cycle of paint baking for auto bodies alters the mechanical properties of aluminum alloys including, possibly, their fracture behavior. In this study, the plasticity and fracture of AA6111 sheet after a 30 min heating cycle at 180 °C is investigated. Uniaxial tension, plane-strain tension and disk compression experiments are performed to assess the plasticity of the material. The results are used to calibrate Yld20004-18p anisotropic yield function. An evolution of the yield locus with plastic deformation is observed. This is represented by evolving the exponent of the yield function, rather than the coefficients themselves. The post-necking hardening curve is identified by numerically simulating a notched-tensile test. The fracture locus is probed using notched-tension, center-hole, and shear experiments. In every case, digital image correlation (DIC) is used to acquire the surface strain fields. In parallel, finite element simulations of the fracture specimens are used to approximate the fracture strains, triaxiality, and Lode angle parameter. Yld2004-18p model shows good overall agreement with the experiments; thus, the fracture locus is probed with this model.

Identification of the post-necking hardening response of rate- and temperature-dependent metals

A method for identifying the material hardening curves past the limit of necking in uniaxial tension and across a range of strain-rates and temperatures in a fully-coupled way is proposed. Experiments on microtubes of 304L stainless steel, which is a rate- and temperature-dependent material, were performed using a custom isothermal testing setup. Digital Image Correlation and Infrared Thermography provided full-field measurements of the strain and temperature during testing. The identification procedure uses a finite element (FE) model of the experiments and the problem is cast as one of mathematical optimization. The corresponding objective function has input parameters that control the post-necking shape of the hardening curves and a scalar output that represents the proximity between the FE predictions of the force-average axial strain response and the experimental data. Since the objective function in not available in closed form and is expensive to evaluate, an efficient optimization procedure that requires a limited number of function evaluations is proposed. The method proposed here is then applied to identifying the post-necking hardening response under different assumptions, starting from a single material curve with no rate and temperature effects included, to a family of curves with both rate and temperature considered in a coupled way. To validate the family of hardening curves identified for 304L stainless steel, a fully-coupled FE model is used to simulate a conventional tension test. This model is shown to be able to reproduce this experiment, including the strain and temperature fields which develop during testing.

Ductile fracture of an Al-Si-Mg die-casting aluminum alloy

Abstract The anisotropic plastic flow and ductile fracture of Al-Si-Mg die-cast aluminum alloy (trade name, Aural2) is investigated using a combination of experiments and analysis. The material is in the form of a 2 mm thick sheet and was received in the –T7 temper. The plasticity of the material is probed using uniaxial tension and plane-strain tension experiments at 3 angles to the material direction, and disc-compression experiments. Good repeatability is obtained in every case. These experiments are used to calibrate the Yld2004-18p anisotropic yield function. This function is flexible enough so that very good fitting of the experiments is obtained. The ductile fracture of the alloy is then probed using notched-tension (with 6 mm and 20 mm notch radii) and central-hole experiments. Each of these specimens is designed so that it provides a different stress triaxiality in the region of rupture, while still requiring only a universal testing machine. Digital Image Correlation is used throughout the experimental work to assess the surface strain fields. However, since the sheet is relatively thick, rupture did not necessarily initiate at the surface. For this reason, fully-3D finite element simulations, using the Yld2004-18p yield function and 3 post-necking hardening models (Swift, Voce and combined) are performed. The surface strains predicted by the simulations depend strongly on the material model adopted. Hence the predicted fracture strains reported here underscore the fact that the plasticity of the material needs to be carefully established before any fracture study is undertaken.

Identification of Post-Necking Hardening Phenomena in Ductile Sheet Metal

A combined theoretical/experimental approach accurately quantifying post-necking hardening phenomena in ductile sheet materials that initially exhibit diffuse necking in tension is presented. The method is based on the minimization of the discrepancy between the internal and the external work in the necking zone during a quasi-static tensile test. The main focus of this paper is on the experimental validation of the method using an independent material test. For this purpose, the uniaxial tube expansion test is used to obtain uniaxial strain hardening behavior beyond the point of maximum uniform strain in a tensile test. The proposed method is used to identify the post-necking hardening behavior of a cold rolled interstitial-free steel sheet. It is demonstrated that commonly adopted phenomenological hardening laws cannot accurately describe all hardening stages. An alternative phenomenological hardening model is presented which enables to disentangle pre- and post-necking hardening behavior. Additionally, the influence of the yield surface on the identified post-necking hardening behavior is scrutinized. The results of the proposed method are compared with the hydraulic bulge test. Unlike the hydraulic bulge test, the proposed method predicts a decreased hardening rate in the post-necking regime which might be associated with probing stage IV hardening. While inconclusive, the discrepancy with the hydraulic bulge test suggests differential work hardening at large plastic strains.

Effects of post-necking hardening behavior and equivalent stress–strain curves on the accuracy of M–K based forming limit diagrams

A rate dependent Taylor type crystal plasticity formulation is employed to study the effects of post-necking hardening behavior on the Marciniak–Kuczynski (M–K) based forming limit diagram predictions. Three uniaxial stress–strain curves, identical up to the necking region, with three different post-necking hardening behaviors are generated using crystal plasticity formulation. The three curves are obtained by curve fitting the simulated uniaxial true stress–strain curve to the experimental curve for the CC AA5754. In the first curve the true stress strain curve saturates after necking while the other two curves demonstrate post-necking hardening. A representative volume element obtained from Electron Backscatter Diffraction (EBSD) data is considered for the analysis. Numerical simulations of forming limit diagrams based on the classical M–K approach are performed with the same initial texture and the three different stress–strain curves. Analysis with the numerical model demonstrates that the post-necking behavior significantly affects the predicted FLDs. A new calibration approach is presented to accurately capture the post-necking behavior of metals. Predicted FLDs with this new approach are compared to the experimental FLD for the CC AA5754. In order to further characterize the process of finding the accurate post-necking hardening behavior, a method based on using equivalent stress–strain curve is presented. It is shown that the FLD, calculated using experimental uniaxial tensile data as input, is comparable with a FLD arising from equivalent stress–strain data.

Identification of Strain Hardening Phenomena in Sheet Metal at Large Plastic Strains

A new experimental/numerical method to identify post-necking strain hardening phenomena in ductile sheet metal is presented. The identification of the post-necking strain hardening behaviour is based on the minimization of the external and the internal work in the necking zone during a tensile test. The proposed method takes the material state and the shape of the whole deforming tensile specimen into account. The post-necking hardening behaviour of a cold rolled interstitial-free steel sheet is identified. A hardening law which enables disentangling pre –and post-necking strain hardening behaviour is presented. The method is experimentally validated using an independent material test. For that purpose, the uniaxial tube expansion test is conducted to obtain uniaxial strain hardening behaviour beyond the point of maximum uniform strain in a tensile test. Finally, the presented method is compared with a hydraulic bulge test.

Characterization of the post-necking strain hardening behavior using the virtual fields method

The present study aims at characterizing the post-necking strain hardening behavior of three sheet metals having different hardening behavior. Standard tensile tests were performed on sheet metal specimens up to fracture and heterogeneous logarithmic strain fields were obtained from a digital image correlation technique. Then, an appropriate elasto-plastic constitutive model was chosen. Von Mises yield criterion under plane stress and isotropic hardening law were considered to retrieve the relationship between stress and strain. The virtual fields method (VFM) was adopted as an inverse method to determine the constitutive parameters by calculating the stress fields from the heterogeneous strain fields. The results show that the choice of a hardening law which can describe the hardening behavior accurately is important to derive the true stress–strain curve. Finally, post-necking hardening behavior was successfully characterized up to the initial stage of localized necking using the VFM with Swift and modified Voce laws.